Fields

Complex Numbers

Definition: Complex numbers are numbers of the form $a+bi$ where $a,b \in \mathbb{R}$ and $i$ is the imaginary unit such that $i^2=-1$. The set of complex numbers is denoted by $\mathbb{C}$.

$$\mathbb{C} = \{a+bi \mid a,b \in \mathbb{R}\}$$

addition and multiplication are defined as follows:

$$\begin{align*} (a+bi)+(c+di) = (a+c)+(b+d)i \\ (a+bi)(c+di) = (ac-bd)+(ad+bc)i \end{align*}$$

The set of real numbers is a subset of the complex numbers, i.e. $\mathbb{R} \subset \mathbb{C}$. The set of imaginary numbers is also a subset of the complex numbers, i.e. $\mathbb{I} \subset \mathbb{C}$.

Euler's formula is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number $x$:

$$e^{ix} = \cos(x) + i \sin(x)$$

where $e$ is Euler's number, the base of natural logarithms, and $i$ is the imaginary unit, which satisfies the equation $i^2 = −1$.

The formula is still valid if $x$ is a complex number. In particular, if $x = \pi$, Euler's formula states that:

$$e^{i\pi} + 1 = 0$$

The complex conjugate of a complex number $z = a + bi$ is given by $\bar{z} = a - bi$. The complex conjugate of a complex number $z$ is denoted by $\bar{z}$.

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#EE501 - Linear Systems Theory at METU